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100 years of Dixon’s identity. (English) Zbl 0795.01009
1991 marks the 100th anniversary of the appearance of a note by Alfred Cardew Dixon proving a famous combinatorial identity. The following slight generalisation also bears Dixon’s name: $$\sum\sb k {a+b \choose a+k} {b+c \choose b+k} {c+a \choose c+k} (-1)\sp k= {(a+b+c)! \over a!b!c!}$$ for nonnegative integers $a,b,c$ and the permitted range of the integer $k$. In the paper we can find three proofs of Dixon’s identity, namely Dixon’s original proof, a second proof using the Lagrange inversion formula, and a third one using WZ pairs. From the paper we can learn some biographical facts about Dixon who was the President of the London Mathematical Society in 1931-33.
01A55Mathematics in the 19th century
05A10Combinatorial functions